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Rule

Negative Exponent Property

If a is a non-zero real number and n is a positive integer, then a raised to the power of - n is equal to 1 over a raised to the power of n.

a^(- n)=1/a^n

It is important to note that the sign of the exponent changes, meaning that the resulting fraction is not simply the reciprocal of the original power.

Proof

To prove this identity, the Zero Exponent Property will be considered. a^0=1, a ≠ 0 In the above equation, 0 will be rewritten as n-n and, after some algebraic manipulation and using the Product of Powers Property, the desired result will be obtained.

a^0=1

WriteDiff

Write as a difference

a^(n-n)=1

WriteSum

Write as a sum

a^(n+(- n))=1

CommutativePropAdd

Commutative Property of Addition

a^(- n +n)=1

SumInExponent

a^(m+n)=a^m*a^n